HILBERT

Constructs an n-by-n Hilbert matrix. A Hilbert matrix is a square matrix with entries H_{i,j} = \frac{1}{i+j-1}. Hilbert matrices are notorious for being highly ill-conditioned.

Excel Usage

=HILBERT(n)

n (int, required): The order of the matrix.

Returns (list[list]): 2D array representing the Hilbert matrix.

Example 1: Hilbert matrix of order 3

Inputs:

n
3

Excel formula:

=HILBERT(3)

Expected output:

Result
1	0.5	0.333333
0.5	0.333333	0.25
0.333333	0.25	0.2

Example 2: Hilbert matrix of order 1

Inputs:

n
1

Excel formula:

=HILBERT(1)

Expected output:

1

Example 3: Hilbert matrix of order 4

Inputs:

n
4

Excel formula:

=HILBERT(4)

Expected output:

Result
1	0.5	0.333333	0.25
0.5	0.333333	0.25	0.2
0.333333	0.25	0.2	0.166667
0.25	0.2	0.166667	0.142857

Example 4: Hilbert matrix from single-cell range input

Inputs:

n
5

Excel formula:

=HILBERT({5})

Expected output:

Result
1	0.5	0.333333	0.25	0.2
0.5	0.333333	0.25	0.2	0.166667
0.333333	0.25	0.2	0.166667	0.142857
0.25	0.2	0.166667	0.142857	0.125
0.2	0.166667	0.142857	0.125	0.111111

Python Code

import numpy as np
from scipy.linalg import hilbert as scipy_hilbert

def hilbert(n):
    """
    Construct a Hilbert matrix.

    See: https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.hilbert.html

    This example function is provided as-is without any representation of accuracy.

    Args:
        n (int): The order of the matrix.

    Returns:
        list[list]: 2D array representing the Hilbert matrix.
    """
    try:
        if isinstance(n, list):
            if not n or not isinstance(n[0], list) or not n[0]:
                return "Error: n must be a scalar or single-cell range"
            n = n[0][0]

        try:
            n_val = int(n)
        except (TypeError, ValueError):
            return "Error: n must be an integer"

        if n_val <= 0:
            return "Error: n must be a positive integer"

        res = scipy_hilbert(n_val)

        return res.tolist()

    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

n *

The order of the matrix.